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A SCALE FOR MEASURING PUPILS' ABILITY TO 
DEMONSTRATE GEOMETRICAL THEOREMS 



J. H. MINNICK 

University of Pennsylvania 



It is desirable that a scale shall be developed for the 
measurement of each of those abilities upon which the mastery 
of geometry is dependent. Among these is the ability to 
demonstrate a theorem when the figure is drawn, and the 
hypothesis and conclusion are given. The chief difficulty in 
developing a scale for measuring this ability is the time element. 
The writing of a complete demonstration requires too much 
of the pupil's time, and the scoring of such a scale is time 
consuming. On a former occasion 1 , the writer used a geometry 
test in which the drawing for a theorem was given and the 
hypothesis and conclusion were stated. The pupil was then 
asked to make any additional drawings necessary to prove 
the theorem. Although the exercises of this test were poorly 
selected, the results were so favorable that it seemed possible* 
by the use of this method, to construct a satisfactory scale. 
This scale would have the advantages of requiring no writing 
on the part of the pupil and of being easily scored. The 
following is a report of an attempt to construct such a scale. 

From various sources eleven exercises, apparently suited 
to the purpose of the scale, were selected. The time available 
for giving the test made it impossible for each pupil to solve 
all the exercises. In order to get a random selection of pupils 
solving each exercise, the following plan was used: Each 
exercise was printed on a separate sheet of paper in the manner 

1 J. H. Minnick. An Investigation of Certain Abilities Fundamental to the Study of 
Geometry. 



102 THE SCHOOL REVIEW 

shown on pages 107-109. The exercises were then assembled in 
groups of three each in such a way that every possible com- 
bination of the exercises was taken, any one combination 
occurring as often as any other. 

The following directions were sent to each teacher who 
gave the tests: 

1. See that each pupil is supplied with a pencil and ruler. 

1. Read to the pupils: I am going to give you some geometry exercises. 
In order that you may all have the same chance, I want you to start at the 
same time. Do not open the set of questions which you are about to receive 
until I give the signal to begin work by tapping on the desk. 

3. Distribute the questions. 

4. Have pupils fill out blanks on the cover sheets of their questions. 

5. Read to the pupils: At the top of each sheet which you have received 
there is a geometrical figure. Below this figure there is a statement of what 
is given and what is to be done. When the signal to begin work is given, 
fold back the cover sheet, read carefully what is given and what is to be 
proved in one of the exercises, and then make any additional drawings 
that are necessary to prove the exercise. Thus, if, in the triangle ABC 
AC = BC [place drawing on the board] and if we are to prove that Z.A = 
LB, we may draw CD to the mid-point of AB. When you have completed 
this exercise proceed in a similar way to do the other exercises. Do not 
write any explanation on your papers and do not prove the exercises. All 
I want to know is whether you can make the correct drawings. You may 
do the exercises in any order you prefer. When you have completed the 
exercises to the best of your ability, hand me your papers. 

6. Give the pupils a chance to ask questions concerning the instructions, 
but do not reveal the contents of the questions by your answers. 

7. Note the time and then give the signal, thus: "Ready," and tap 
on the desk with your pencil. 

8. Do not permit talking in the class until all papers have been returned 
to you. 

9. In the case of any irregularity on the part of any pupil during the 
test, make a note on the cover sheet of his questions indicating the exact 
nature of the irregularity. 

10. As the pupils hand you their papers, note the amount of time 
required by each, and record it on the upper right-hand corner of the cover 
sheet of his or her paper. 



PUPILS' ABILITY TO DEMONSTRATE THEOREMS 



103 



The tests were given in thirty high schools distributed 
throughout the country and ranging in size from a few hundred 
pupils to several thousand. Each of the eleven exercises was 
solved by at least seven hundred pupils. These pupils had 
completed either the first two books of plane geometry or all 
of plane geometry. 



TABLE I 

Percentage of Pupils Solving Each Exercise Correctly 



Number of 
Pupils 



Exercises 



50 
ICO 

150 
200 
250 
300 

35° 
400 
450 
500 

55° 
600 
650 

700 



42 
37 
34 
34 
33 
35 
38 
38 
41 
41 
41 
42 
42 
42 



76 

70 

7i 

72 

75 
76 
77 
77 
77 
79 
81 
81 
82 
82 



20 
28 
33 
37 
42 
45 
49 
46 
46 
48 
49 
49 
49 
49 



42 
42 
38 
38 
37 
37 
37 
38 
36 
35 
3° 
29 
28 
28 



68 
62 

63 
64 
64 
66 

67 
68 
66 
66 
66 
66 

67 
68 



28 
20 
22 
26 
28 

30 
32 
33 
35 
34 
32 
32 
3i 
3i 



68 
60 
60 
64 
60 
61 
64 
64 
62 
60 
59 
58 
58 
57 



14 
12 

l 3 

'5 

12 

J 5 

14 
H 
14 
14 
13 
13 



40 
41 
41 
44 
45 
45 
45 
43 
44 
42 
42 

41 
40 

39 



16 

24 
40 
46 

43 
41 
36 

32 
28 

25 
23 
22 

19 

20 



46 

47 
36 
36 
36 
37 
38 
36 
37 
36 
36 
35 
34 
34 



In order to secure uniformity of marking, all papers were 
marked by the author. Any drawing which made a proof 
possible was usually counted as a correct solution. If, how- 
ever, in addition to the correct lines, unnecessary lines were 
drawn, the solution was marked as incorrect. Also, if the 
proof resulting from a drawing seemed to be too difficult for 
high-school pupils, those pupils making the drawing were asked 
to complete the proof. If the majority could not do so, the 
drawing was not accepted as a correct solution. A careful 



io 4 THE SCHOOL REVIEW 

record of all acceptable drawings has been kept and will be 
included in the "Directions for Scoring Papers 1 ". 

Table I gives the percentage of pupils solving each exercise 
correctly. The table is arranged in a cumulative way, each 
new line including the data for fifty additional pupils. Thus 
of the first fifty pupils solving Exercise i, 42 per cent got it 
correct; of the first hundred solving Exercise i, 37 per cent 
got it correct, etc. The table shows that the number of pupils 
tested has not been sufficient to eliminate completely individ- 
ual variation. However, the percentages are fairly constant, 
and it is not probable that the addition of data from more 
pupils would vary the order of difficulty or seriously change 
the weighting of the exercises. 




Very poor A M Very good 

Fig. 1 

In weighting the exercises we have assumed that the 
distribution of pupils according to the ability in question will 
result in the normal frequency-curve as shown in Fig. 1. The 
direction from left to right on CD will be considered positive. 
The line MN represents the median. If AP is drawn so that 
AMNP is one-fourth the entire surface under the curve, then 
AM is known as the P. E. (possible error) . 

Reference to Table I shows that Exercise 2 is the easiest 
of the eleven exercises used. The score for this exercise is 82. 
It is, therefore, 82 — 50, or 32 per cent too easy for the median 
pupil. Converting this difference into P. E. 2 values, we find 
that Exercise 2 falls at — 1.357 P. E.; that is, 1.357 P. E. to the 

1 Information concerning the prices of tests and directions for giving and scoring the 
tests may be had by addressing J. H. Minnick, College Hall, University of Pennsylvania, 
Philadelphia, Pennsylvania. 

2 Table XIII of Trabue's Completion Test Language Scale. 



PUPILS' ABILITY TO DEMONSTRATE THEOREMS 105 

left of the median. Since our zero-point must be arbitrary, 
we may select it so as to give convenient numbers for values 
of the various exercises. Hence we have selected —2.357 P. E. 
as our zero-point. This makes Exercise 2 fall at 1 P. E. above 
the zero-point. 

Table II indicates the method of determining the value of 
each exercise. Thus the first line indicates that 42 per cent 
of the pupils solved Exercise 1 correctly, that the exercise is 8 

TABLE II 

Values Assigned to Each Exercise 



Exercise 


Per Cent 
Correct 


Difference 

between 

50 Per Cent 

and Score 


P. E. 

Value 


Distance 

in P. E. 

above 

Zero-Point 


Value 




42.O 
82.O 

49-3 
27.9 
67.7 
3*3 
56.7 
l 3-3 
39-3 
20.0 

34-o 


+ 8.0 

—32.O 

+ 0.7 
+ 22. 1 
—17.7 

+ 18.7 
-6.7 

+ 36.7 
+ IO.7 
+30.0 
+ I6.0 


+ O.299 

—'•357 
+0.026 
+0.869 
-0.681 
+0.723 
— 0.250 
+ 1.649 

+0.403 
+ 1.248 
+0.612 


2.656 

1 .OOO 

2 383 
3.226 
I.676 
3.080 
2.IO7 
4.006 
2.760 
3.605 
2.969 


27 
IO 


2 


7 


24 

3 2 
17 
3i 
21 




C 


6 


7 


8 


40 
28 


q 


IO 


36 
3° 


II 





per cent too difficult for the median pupil and is, therefore, 
0.299 P- E. above the median or 2.656 P. E. above our arbitrary 
zero-point, and that the value assigned to Exercise 1 is 27. 
In order to avoid the decimal point in the values assigned, 
0.1 P. E. has been taken as the unit. The values are then 
obtained by moving the decimal point one place to the right 
in the next to the last column of the table. 

Fig. 2 is the linear projection of the eleven exercises. 
From these exercises it was necessary to select those which 
should constitute the scale. When the tests were given, a 



io6 



THE SCHOOL REVIEW 



record was kept of the time required by each pupil to do the 
three exercises assigned to him. This time varied from 5 to 
30 minutes. The average time spent by a pupil was about 1 8 
minutes. It, therefore, seemed that five exercises were sufficient 
for the 15 or 20 minutes 1 which the pupils will be given to 
work on the final scale. An examination of Fig. 2 shows that 



Exerolses- 



1 r.E. 

1 — 



2 ».«. 



t 

Fig. 2 



3 P.E. 



19 11 6 4 



■"I— 
10 



4 P.E. 

>l 



Exercises 2, 5, 3, and 6 are distributed along the scale at almost 
equal intervals. The interval between 6 and 10 is somewhat 
smaller, and that between 6 and 8 somewhat larger than the 
intervals by which 2, 5, 3, and 6 are separated. Exercise 10 
occurs as a theorem in some texts, and the results from the 
schools in which the tests were given were clearly affected by 
this fact. Hence Exercise 8 was selected as the fifth exercise 
of our scale. 

TABLE III 
Order of Difficulty for the Eight Largest Schools 



Schools 


Order 


1 


2 


3 


4 


5 


A 


2 
2 
2 
2 

2 

5 

2 

2 


5 
3 
5 

5 
5 
2 

3 

5 


3 
5 
3 
6 

3 
3 
5 
3 


6 
6 
6 

3 
6 
6 
6 
6 


8 


B 


8 


C 


8 


D 


8 


E 


8 


F 


8 


G 


8 


H 


8 







The selection of these five exercises is also justified by the 
fact that two fall above, two below, and one almost at the 



■This time is to be determined experimentally during the year 1919-20. 



PUPILS' ABILITY TO DEMONSTRATE THEOREMS 107 

median. Also, if we examine Table I, we find that these 
exercises have been given to enough pupils to give almost 
constant results. The order of difficulty of these five exercises 
for students completing the first two books of geometry and 
for pupils completing all of plane geometry has been carefully 
examined and has been found to be identical in the two cases. 
Table III shows the order of difficulty for the eight largest 
schools. Thus for School A the array of exercises from the 
easiest to the most difficult is 2, 5, 3, 6, 8. Four of the eight 
schools have the same order of difficulty. In each of the other 
schools there is one and only one reversal of the order. In no 
case is an exercise removed more than one place from the order 
given by all of the schools taken together. 

Exercises 2, 5, 3, 6, 8, will constitute our test and will be 
numbered I, II, III, IV, V, respectively. The exercises in full 
follow: 




Given: The circle whose center is O, arc AB = arc BC, BM is 
perpendicular to AO, BN is perpendicular to OC. Make any additional 
drawings that are necessary to prove that BM = BN. 



io8 



THE SCHOOL REVIEW 

ll 



-B 




F 

Given: AB is parallel to CD, and lines GE and FE meet in E. 
Make any additional drawings that are necessary to prove that 
Z2= Zi+ Z3- 



HI 




Given: Triangle ABC inscribed in a circle whose center is 0, and OD 
perpendicular to CB. 

Make any additional drawings that are necessary to prove that Zi= Z2. 



PUPILS' ABILITY TO DEMONSTRATE THEOREMS 



109 




Given: The triangle ABC, AC = BC, D is any point on AB, DE is 
perpendicular to BC, DG is perpendicular to AC. 

Make any additional drawings necessary to prove that DG + DE = AF. 




Given: The circle whose center is 0, chord AC = chord BD, AC and BD 
intersect in P, and the line OP is drawn. 

Make any additional drawings that are necessary to prove that Z 1 = /.1.